KS3 Maths · Year 8 · Chapter 5: Applications of graphs · Lesson 1

Step graphs

Year 8About 60 minutesNo calculator47 marks

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Date

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Quick questions to start.

A1
Needed today
How many minutes are there in $3.5$ hours?
[1 mark]
Answer
A2
Needed today
Work out $4 \times$ £ $2.35$ .
[1 mark]
Answer
A3
From chapter 4
Work out the mean of $6$ , $9$ , $10$ and $15$ .
[1 mark]
Answer
A4
From chapter 3
Work out the size of one exterior angle of a regular hexagon.
[1 mark]
Answer

Example, then your turn

Year 8

On these graphs a filled circle is included in its step and an open circle is not. Your teacher works through each example with you. Then try the one beside it.

Example 1

The step graph shows the charges at a car park. (a) Work out the cost of parking for $40$ minutes, for exactly $2$ hours, and for $4$ hours $20$ minutes. (b) Jo has £ $5$ . What is the longest time she can park? (c) Sam paid £ $3.50$ . Write down the range of times he could have parked for.

Hint To read a cost, go up from the time to the step above it. To read backwards, go across from the cost to its step: the step runs from its open circle to its filled circle.

B1
The step graph shows the charge for a delivery. (a) Work out the charge for a delivery of $8$ km, of exactly $10$ km, and of $25$ km. (b) A delivery cost £ $10$ . Write down the range of distances it could have been.
[5 marks]

Example 2

The graph shows bus fares by age. Children under $5$ travel free. Here the filled circle is at the START of each step. Work out the fare for a person aged (a) $11$ , (b) $15$ , (c) $16$ .

Hint At age

11

there is an open circle on the £

1

step and a filled circle on the £

1.50

step. The filled circle is the one that counts.

B2
The graph shows the price of a bowling game by age. The filled circle is at the start of each step. Work out the price for a person aged (a) $9$ , (b) $10$ , (c) $16$ .
[3 marks]
Answer(a) £ $\dots$ (b) £ $\dots$ (c) £ $\dots$

Example 3

A courier charges £ $3$ for a parcel up to $1$ kg, £ $4.50$ for more than $1$ kg up to $2$ kg, £ $5.50$ for more than $2$ kg up to $3$ kg, and £ $7$ for more than $3$ kg up to $5$ kg. Draw a step graph to show these charges.

Hint Draw one flat line for each charge. Put an open circle at the start of each line and a filled circle at the end.

B3
A boat costs £ $8$ to hire for up to $1$ hour, £ $14$ for more than $1$ hour up to $2$ hours, and £ $18$ for more than $2$ hours up to $4$ hours. Draw a step graph on the grid to show these charges.
[3 marks]

Practice

Year 8

C1
The graph shows a library's fines for overdue books. Lee returns four books. Two are $4$ days overdue, one is $7$ days overdue and one is $12$ days overdue. Work out his total fine.
[3 marks]
Answer£ $\dots$
C2
The graph shows a waiter's pay for one day at the weekend. Jaya works $3$ hours on Saturday and $5$ hours on Sunday. She is also given $10%$ of her pay as tips. How much does she get altogether?
[3 marks]
Answer£ $\dots$

C3
A parcel firm charges £ $6.50$ to send a parcel of up to $2$ kg. For a parcel over $2$ kg, it charges £ $1.20$ more for each extra $500$ g, or part of $500$ g. Work out the cost of sending a parcel of $3.3$ kg.
[3 marks]
Answer£ $\dots$
C4
Which of these gives a step graph? Give a reason. A: the distance a car travels at a steady speed. B: the cost of hiring a bike at £ $5$ for each hour or part of an hour. C: the temperature of a cup of tea as it cools.
[2 marks]
Answer $\dots\dots$

Exam-style questions

GCSE Foundation

D1
The step graph shows the charges at a station car park.
(a)
Write down the cost of parking for $1$ hour.
[1 mark]
Answer£ $\dots$
(b)
Rosa parks from 10:35 to 13:05. Work out the cost.
[2 marks]
Answer£ $\dots$
(c)
Tom parks twice: once for $45$ minutes and once for $3$ hours $20$ minutes. He pays with a £ $10$ note. How much change should he get?
[2 marks]
Answer£ $\dots$
(d)
Work out the cost per hour of parking for exactly $4$ hours and for exactly $6$ hours. Which is cheaper per hour?
[2 marks]
Total for D1: 7 marks

D2
Ella says, "Parking for exactly $2$ hours costs £ $6$ , because the price goes up at $2$ hours."
A car park charges £ $4$ for more than $1$ hour up to and including $2$ hours, and £ $6$ for more than $2$ hours up to $4$ hours. Is Ella correct? Give a reason for your answer.
[2 marks]
D3
A taxi charges £ $3$ for a journey up to $1$ mile. For each extra mile, or part of a mile, it charges £ $2$ more. The charges go up to $5$ miles.
(a)
Draw a step graph on the grid to show the charges for journeys up to $5$ miles.
[3 marks]
(b)
Work out the fare for a journey of $3.4$ miles.
[1 mark]
Answer£ $\dots$
Total for D3: 4 marks
D4
Company A charges for parcels using the table. Company B charges £ $2.30$ for each kilogram, or part of a kilogram. Megan wants to send a parcel with a mass of $3.6$ kg. Which company is cheaper, and by how much? You must show your working.
[4 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Car park A charges £ $1.80$ for each hour or part of an hour. Car park B charges £ $3$ for up to $2$ hours, £ $6$ for more than $2$ hours up to $5$ hours, and £ $9$ for more than $5$ hours up to $12$ hours. For which stays of up to $6$ hours is car park B cheaper? Give your answer as ranges of time.
[4 marks]
Hint 1 · What to try
Work out both costs for each hour: a stay of up to $1$ hour, more than $1$ up to $2$ hours, and so on.
Hint 2 · The first line
Up to $1$ hour: A £ $1.80$ , B £ $3$ . More than $1$ up to $2$ hours: A £ $3.60$ , B £ $3$ .
Hint 3 · The full method
Continue: $2$ – $3$ h: A £ $5.40$ , B £ $6$ ; $3$ – $4$ h: A £ $7.20$ , B £ $6$ ; $4$ – $5$ h: A £ $9$ , B £ $6$ ; $5$ – $6$ h: A £ $10.80$ , B £ $9$ . So B is cheaper for more than $1$ up to $2$ hours, and for more than $3$ up to $6$ hours.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can read a cost from a step graph, and read a step graph backwards.
I can use open and filled circles to tell which end of a step is included.
I can draw a step graph from a list of charges.
I can work out charges such as "each extra $500$ g or part of $500$ g".

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